Abstract
The Gottesman-Kitaev-Preskill (GKP) error-correcting code encodes a finite-dimensional logical space in one or more bosonic modes, and has recently been demonstrated in trapped ions and superconducting microwave cavities. In this work we introduce a new subsystem decomposition for GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to quantum stabilizer codes. The decomposition has the defining property that a partial trace over the nonlogical stabilizer subsystem is equivalent to an ideal decoding of the logical state, distinguishing it from previous GKP subsystem decompositions. We describe how to decompose arbitrary states across the subsystem decomposition using a set of transformations that move between the decompositions of different GKP codes. Besides providing a convenient theoretical view on GKP codes, such a decomposition is also of practical use. We use the stabilizer subsystem decomposition to efficiently simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis simulations, we are able to consider essentially arbitrarily large photon numbers for realistic noise channels, such as loss and dephasing.
- Received 9 December 2022
- Revised 28 July 2023
- Accepted 10 January 2024
DOI:https://doi.org/10.1103/PRXQuantum.5.010331
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
The promise of a general-purpose quantum computer presents both an exciting scientific opportunity and a daunting engineering challenge because of the low error rates that need to be maintained. One solution involves the Gottesman-Kitaev-Preskill (GKP) code, which uses quantum harmonic oscillators to protect the information in the quantum computer. In this work, we develop a novel theoretical framework to describe the operation of the GKP code. In doing so, we provide both an intuitive theoretical picture of the code and an efficient numerical method for simulating noise processes.
The key mathematical tool that we utilize is called a subsystem decomposition, which was recently proposed as a way to describe GKP codes. However, here we alter the details of the decomposition to reflect the error-correction capabilities of the GKP code. Not only does this provide an intuitive picture of the error-correction process, but it also allows us to develop a novel numerical method to simulate the effects of noise on the GKP code. Our numerical method has the advantage that it can be applied both to realistic sources of noise and to GKP codestates of arbitrarily high quality—something that existing numerical techniques cannot do.
Some of the next steps include applying our numerical techniques to a wider set of noise channels and improving its efficiency for calculations involving multiple GKP modes.
